#151892

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#346
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

322 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6)(7,19,21,33)(8,15)(9,27)(10,17)(11,18)(12,14)(13,20,34,24)(16,28)(22,49,47,43)(23,44,48,58)(25,41)(26,39)(29,55)(30,38)(31,42)(32,37)(35,50,54,45)(36,46,59,52)(40,56)(51,66,65,63)(53,62)(57,61)(60,67,68,64);(2,4)(5,7,9,22,10,23,29,51,11,24,30,50,31,52,57,67,12,21,28,47,32,48,56,65,8,20,25,45,26,46,53,64)(6,13,16,35,17,36,40,60,18,33,41,49,42,58,62,66,14,34,27,54,37,59,55,68,15,19,38,43,39,44,61,63);(6,14)(7,20,21,24)(9,28)(13,33,34,19)(15,18)(17,37)(22,50,47,45)(23,46,48,52)(25,30)(29,56)(35,43,54,49)(36,58,59,44)(39,42)(51,67,65,64)(53,57)(60,63,68,66)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fi,c^b=ch,e^b=ei,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322699, #72323078
maximal normal subgroup of	#72322699, #72323078
maximal quotient of	#72322700

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59088);