#7197517

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
prime radical	#24218	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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)(17,18,23,30,22,26,21,27)(19,28,31,24,29,20,25,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,19,22,29)(18,24,26,32)(20,30,28,27)(21,31,23,25);(2,6)(5,7)(9,11,13,10)(12,15,16,14)(17,19)(18,20)(21,25)(22,29)(23,31)(24,27)(26,28)(30,32);(3,8)(5,7)(10,11)(14,15)(17,20,23,32,22,28,21,24)(18,25,30,19,26,31,27,29);(9,12,13,16)(10,14,11,15)(18,26)(19,29)(25,31)(27,30)

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,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, 7104713);