#230453

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

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 ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

568 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137649);