#7533682

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2017 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)(13,14,19,24)(15,25,28,22)(16,21,29,37)(17,20,33,36)(18,34,26,23)(27,38,42,40)(30,43,32,35)(31,44,41,39);(1,2)(3,4)(7,12)(9,11)(13,15,26,33,19,28,18,17)(14,20,34,22,24,36,23,25)(16,27,41,30,29,42,31,32)(21,35,44,40,37,43,39,38);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,27)(17,32)(18,31)(19,29)(20,35)(22,40)(23,39)(24,37)(25,38)(26,41)(28,42)(30,33)(34,44)(36,43);(2,4)(6,10)(9,11)(16,29)(21,37)(27,42)(30,32)(31,41)(35,43)(38,40)(39,44);(13,17,18,28,19,33,26,15)(14,22,23,20,24,25,34,36)(16,30,31,27,29,32,41,42)(21,38,39,43,37,40,44,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,e^2=f,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^c=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, 7440878);