#190429

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2283 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dh,g^b=gh,d^c=di,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, 97625);