#7010900

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1339 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6918096);