#414525

order	512
nilpotency class	3
isoclinism	isoclinic class of #395154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=ef,c^2=ef,d^2=gh,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=di,d^c=di

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, 321721);