#454957

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1464 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)(13,14,17,23,19,25,18,24)(15,20,27,22,29,37,28,36)(16,30,32,40,34,39,33,21)(26,41,42,44,43,38,31,35);(6,9)(10,12)(13,15)(14,20)(16,31)(17,27)(18,28)(19,29)(21,38)(22,23)(24,36)(25,37)(26,32)(30,35)(33,43)(34,42)(39,44)(40,41);(2,4)(7,11)(10,12)(13,16,19,34)(14,21,25,40)(15,26,29,43)(17,33,18,32)(20,35,37,44)(22,38,36,41)(23,39,24,30)(27,31,28,42);(5,7,8,11)(6,10,9,12)(13,15)(14,22)(16,26)(17,28)(18,27)(19,29)(20,23)(21,38)(24,37)(25,36)(30,44)(31,32)(33,42)(34,43)(35,39)(40,41)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362153);