#7223456

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200002
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	#2541	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1403 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)(3,6)(4,7)(5,8)(11,15)(14,16)(17,18,20,22)(19,25,26,32)(21,29,28,24)(23,30,31,27);(1,3,6,2)(4,7,5,8)(9,10)(11,16)(12,13)(14,15)(18,22)(23,31)(24,29)(25,32);(10,13)(14,16)(17,19,20,26)(18,23,22,31)(21,30,28,27)(24,32,29,25);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(18,22)(23,31)(24,29)(25,32);(1,5,6,4)(2,7,3,8)(10,13)(14,16)(18,22)(23,31)(24,29)(25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7130652);