#443778

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
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

1078 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:

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

Polycyclic presentation:

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

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, 350974);