#7145940

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	#45267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2169 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)(7,11)(10,12)(13,14,19,25)(15,26,29,21)(16,23,17,32)(18,24,20,35)(22,34,39,31)(27,37,33,40)(28,42,30,38)(36,43,41,44);(2,3)(6,9)(7,11)(13,15)(14,21)(16,27)(17,33)(18,28)(19,29)(20,30)(22,36)(23,40)(24,38)(25,26)(31,43)(32,37)(34,44)(35,42)(39,41);(5,6)(7,10)(8,9)(11,12)(13,16,19,17)(14,22,25,39)(15,27,29,33)(18,31,20,34)(21,36,26,41)(23,24,32,35)(28,43,30,44)(37,42,40,38);(1,3,4,2)(6,9)(10,12)(13,17,19,16)(14,23,25,32)(15,27,29,33)(18,31,20,34)(21,37,26,40)(22,24,39,35)(28,44,30,43)(36,42,41,38);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,24,25,35)(15,28,29,30)(16,31,17,34)(21,38,26,42)(22,32,39,23)(27,43,33,44)(36,37,41,40)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,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, 7053136);