#7186583

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1784 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)(21,22,26,30)(23,31,27,28)(24,33,35,32)(25,29,36,34);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,26,27)(22,28,30,31)(24,34,35,29)(25,33,36,32);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,27,34,26,35,23,29)(22,25,31,33,30,36,28,32);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,25)(22,29)(23,32)(24,31)(26,36)(27,33)(28,35)(30,34);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,30)(25,36)(28,31)(32,33)

Polycyclic presentation:

2^9|a^2=f,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,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314172
maximal normal subgroup of	#72314172

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7093779);