#7511109

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,4)(5,7)(6,8)(9,10,12,13)(11,15,16,14)(17,18,24,26)(19,22,21,23)(20,25,31,32)(27,30,28,29);(1,3)(2,6)(4,7)(5,8)(17,19,24,21)(18,22,26,23)(20,25,31,32)(27,29,28,30);(10,13)(14,15)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32);(2,7)(3,8)(9,11,12,16)(10,14,13,15)(17,21,24,19)(18,22,26,23)(20,28,31,27)(25,30,32,29);(1,4,5,6)(2,3,7,8)(17,22,24,23)(18,19,26,21)(20,29,31,30)(25,28,32,27)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,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, 7418305);