#358209

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1520 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)(7,10)(8,9)(11,12)(13,14,18,22,19,23,17,31)(15,20,25,36,26,37,32,35)(16,27,30,21,29,40,33,39)(24,38,42,43,41,44,28,34);(6,9)(10,12)(13,15)(14,20)(16,28)(17,32)(18,25)(19,26)(21,38)(22,36)(23,37)(24,30)(27,34)(29,42)(31,35)(33,41)(39,44)(40,43);(2,4)(13,16)(14,21)(15,24)(17,33)(18,30)(19,29)(20,34)(22,40)(23,39)(25,42)(26,41)(27,31)(28,32)(35,44)(36,38)(37,43);(5,7,8,11)(6,10,9,12)(16,29)(21,39)(24,41)(27,40)(28,42)(30,33)(34,43)(38,44)

Polycyclic presentation:

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

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, 265405);