#474171

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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 ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1040 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:

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

Polycyclic presentation:

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

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, 381367);