#6669740

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	C₂×#3307	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1829 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)(3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,25,28)(23,27,32,36)(24,29,34,26)(30,31,33,35);(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,25,32)(22,27,28,36)(24,31,34,35)(26,30,29,33);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,28)(23,30)(26,29)(27,35)(31,36)(32,33);(6,12)(10,13)(11,18)(17,19)(21,24)(22,29)(23,31)(25,34)(26,28)(27,33)(30,36)(32,35);(7,15)(10,13)(14,20)(17,19)(23,30)(24,34)(26,29)(27,31)(32,33)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308696, #72308867
maximal normal subgroup of	#72308696, #72308867

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6576936);