#444405

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 351601);