#427550

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1522 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)(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,25,29,32)(21,30,26,24)(23,31,27,28);(1,2)(3,4)(17,19,26,31,22,29,21,28)(18,23,30,32,20,27,24,25);(2,4)(10,13)(14,16)(17,20,22,18)(19,27,29,23)(21,30,26,24)(25,28,32,31);(6,8)(9,11,12,15)(10,14,13,16)(19,28)(21,26)(23,25)(24,30)(27,32)(29,31)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=dei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334746);