#213395

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

556 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)(3,8)(4,6)(5,7)(9,10,14,15,13,16,11,12)(19,25)(20,26)(22,28)(24,29)(27,31)(30,32);(1,3,5,4)(2,6,7,8)(9,11,13,14)(10,15,16,12)(17,18,21,23)(19,22,27,30)(20,24,26,29)(25,28,31,32);(17,19,26,31,21,27,20,25)(18,22,29,32,23,30,24,28);(1,4,5,3)(2,6,7,8)(9,12,11,16,13,15,14,10)(18,23)(19,27)(24,29)(25,31)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=fh,d^2=eg,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dhi,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120591);