#141176

order	512
nilpotency class	4
isoclinism	isoclinic class of #125496
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

420 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=ei,d^2=gh,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,d^c=di,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48372);