#7169115

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	#42508	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,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.

Relationships

maximal subgroup of	#72313815, #72313852, #72313939
maximal normal subgroup of	#72313815, #72313852, #72313939

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7076311);