#420714

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
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 ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1358 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)(9,10,13,12)(11,15,16,14)(17,18,19,25,21,26,22,23)(20,28,30,24,29,31,27,32);(1,2)(3,4)(9,11,13,16)(10,14,12,15)(17,19,21,22)(18,23,26,25)(20,27,29,30)(24,31,32,28);(2,4)(9,10,13,12)(11,14,16,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32);(6,8)(9,12,13,10)(11,14,16,15)(17,19,21,22)(18,25,26,23)(20,27,29,30)(24,28,32,31)

Polycyclic presentation:

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

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, 327910);