#464852

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

1202 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=e,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, 372048);