#390478

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1089 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,23,19,24,17,22)(15,20,26,34,27,35,32,40)(16,21,31,39,29,37,30,38)(25,33,28,36,41,43,42,44);(5,7,9,6)(8,11,12,10)(13,15)(14,20)(16,28)(17,32)(18,26)(19,27)(21,36)(22,40)(23,34)(24,35)(25,30)(29,42)(31,41)(33,38)(37,44)(39,43);(2,4)(5,8,9,12)(6,10,7,11)(13,16)(14,21)(15,25,27,41)(17,30)(18,31)(19,29)(20,33,35,43)(22,38)(23,39)(24,37)(26,28,32,42)(34,36,40,44);(2,4)(6,7)(10,11)(16,29)(21,37)(25,41)(28,42)(30,31)(33,43)(36,44)(38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=egh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,d^c=di

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, 297674);