#128230

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1744
prime radical	C₂³	#431
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

234 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=dh,d^2=g,e^2=h,h^2=i,b^a=bd,f^a=fg,c^b=ci,e^b=eg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35426);