#465219

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

912 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eg,c^2=fi,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,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, 372415);