#7324905

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	#52956	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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)(3,7)(4,6)(5,8)(17,18,23,28)(19,30,31,20)(21,26,25,32)(22,29,24,27);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,20)(21,24)(22,25)(23,31)(26,29)(27,32)(28,30);(9,11)(10,13)(12,16)(14,15)(17,20,22,32,23,30,24,26)(18,25,27,31,28,21,29,19);(3,8)(5,7)(10,14)(11,16)(17,21)(18,26)(19,22)(20,27)(23,25)(24,31)(28,32)(29,30);(2,6)(5,7)(10,14)(13,15)(17,22,23,24)(18,27,28,29)(19,21,31,25)(20,26,30,32)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7232101);