#7182692

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
prime radical	#24662	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1751 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)(8,16)(11,17)(15,20)(18,19)(21,22,28,25)(23,24,33,35)(26,30,27,36)(29,31,34,32);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,26,31,28,33,27,32)(22,29,36,35,25,34,30,24);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,23)(25,33)(26,29)(27,34)(28,35)(30,32)(31,36);(7,14)(10,13)(15,20)(18,19)(21,25,28,22)(23,24,33,35)(26,36,27,30)(29,31,34,32);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,28,27)(22,30,25,36)(23,31,33,32)(24,34,35,29)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7089888);