#457540

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

873 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=ei,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=gi,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, 364736);