#383953

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291149);