#7197533

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7104729);