#457421

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=ei,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,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, 364617);