#7192995

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	#26943	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1804 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,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	#72257058, #72257108, #72257166
maximal normal subgroup of	#72257058, #72257108, #72257166
maximal quotient of	#72264901

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7100191);