#477128

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
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

846 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)(6,8)(10,12)(13,14,17,22)(15,19,27,38)(16,30,32,20)(18,23,35,24)(21,33,41,34)(25,37,43,36)(26,40,31,44)(28,39,29,42);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,31)(17,27)(18,29)(20,40)(21,36)(22,38)(23,42)(24,39)(25,33)(26,32)(28,35)(30,44)(34,43)(37,41);(2,3)(5,7,9,11)(6,10,8,12)(13,16)(14,20)(15,25)(17,32)(18,34)(19,36)(21,24)(22,30)(23,41)(26,28)(27,43)(29,31)(33,35)(37,38)(39,44)(40,42);(2,3)(5,8,9,6)(7,10,11,12)(13,16)(14,21)(15,26)(17,32)(18,33)(19,37)(20,24)(22,41)(23,30)(25,28)(27,31)(29,43)(34,35)(36,38)(39,44)(40,42)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384324);