#6527047

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35534	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2062 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)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(22,28)(23,30)(25,26)(29,35)(31,36)(33,34);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,22,25,28,27,30,26,23)(24,29,33,35,32,36,34,31);(7,13)(8,15)(10,16)(11,18)(21,23,26,30,27,28,25,22)(24,31,34,36,32,35,33,29);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(21,24)(22,29)(23,31)(25,33)(26,34)(27,32)(28,35)(30,36);(7,13)(10,16)(14,19)(17,20)(24,32)(29,36)(31,35)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6434243);