#6996919

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
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	C₂×#2011	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2031 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256902, #72256915, #72256963
maximal normal subgroup of	#72256902, #72256915, #72256963
maximal quotient of	#72308317

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6904115);