#6524941

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500060
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×#3214	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1922 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)(7,10)(8,12)(9,11)(13,17)(14,16)(15,18)(19,20)(21,22,26,35,25,30,24,29)(23,31,33,28,32,27,34,36);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,32)(22,27,30,31)(24,34,26,33)(28,35,36,29);(6,11)(7,13)(8,15)(16,20)(21,23)(22,28)(24,33)(25,32)(26,34)(27,29)(30,36)(31,35);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,16,17,20)(23,32)(27,31)(28,36)(33,34);(7,13)(10,17)(14,19)(16,20)(21,24,25,26)(22,29,30,35)(23,33,32,34)(27,36,31,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi

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, 6432137);