#224064

order	512
nilpotency class	3
isoclinism	isoclinic class of #196038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=h,d^2=ei,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,g^c=gh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 131260);