#7193395

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1815 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)(22,28)(23,33)(25,26)(29,35)(30,36)(31,32);(5,7,9,14)(6,10,13,11)(8,17,18,15)(12,20,16,19)(21,22,27,32)(23,30,33,29)(24,35,34,36)(25,28,26,31);(1,2,3,4)(5,7)(6,11)(8,15)(9,14)(10,13)(12,20)(16,19)(17,18)(21,23,27,33)(22,29,32,30)(24,26,34,25)(28,36,31,35);(7,14)(10,11)(15,17)(19,20)(21,24)(22,30)(23,25)(26,33)(27,34)(28,36)(29,32)(31,35);(5,8,9,18)(6,12,13,16)(7,15,14,17)(10,19,11,20)(21,25,27,26)(22,31,32,28)(23,34,33,24)(29,36,30,35)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308532, #72308571
maximal normal subgroup of	#72308532, #72308571

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7100591);