#7006459

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2012 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,19,32,17,20,18,23)(15,24,31,21,26,37,27,34)(16,28,33,36,29,22,30,43)(25,39,44,42,40,38,41,35);(2,3)(5,6)(7,11)(8,10)(9,12)(14,20)(16,29)(21,34)(23,32)(24,37)(25,40)(30,33)(41,44);(5,7)(6,9)(8,12)(10,11)(13,15,17,26)(14,21,20,34)(16,25,29,40)(18,31,19,27)(22,35,28,42)(23,37,32,24)(30,44,33,41)(36,39,43,38);(1,3,4,2)(7,12)(9,11)(13,16,17,29)(14,22,20,28)(15,25,26,40)(18,30,19,33)(21,35,34,42)(23,36,32,43)(24,38,37,39)(27,41,31,44);(6,10)(9,11)(16,29)(22,28)(25,40)(30,33)(35,42)(36,43)(38,39)(41,44)

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6913655);