#412141

order	512
nilpotency class	3
isoclinism	isoclinic class of #394797
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3153	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319337);