#7533714

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1975 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7440910);