#458604

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
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

1070 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)(17,18,22,27)(19,28,31,32)(20,26,21,25)(23,30,24,29);(1,2)(3,4)(9,11,13,12)(10,14,15,16)(17,19,22,31)(18,23,27,24)(20,30,21,29)(25,32,26,28);(2,4)(9,12,13,11)(10,14,15,16)(17,19,21,30,22,31,20,29)(18,23,26,32,27,24,25,28);(6,8)(17,19,22,31)(18,24,27,23)(20,30,21,29)(25,28,26,32)

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=fi,d^2=i,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,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^d=ei,f^d=fi,g^d=gi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365800);