#7186514

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#26601	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2,4,3)(8,16)(11,17)(15,20)(18,19)(21,22,26,29)(23,27,25,34)(24,31,33,35)(28,30,36,32);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,26,25)(22,27,29,34)(24,32,33,30)(28,35,36,31);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,25,32,26,33,23,30)(22,28,27,31,29,36,34,35);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,23,26,25)(22,27,29,34)(24,30,33,32)(28,31,36,35);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,29)(27,34)(28,36)(31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257067, #72257105, #72257166
maximal normal subgroup of	#72257067, #72257105, #72257166
maximal quotient of	#72265576

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7093710);