#7304747

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300948
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 ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1719 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284745, #72285261
maximal normal subgroup of	#72284745, #72285261

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7211943);