#378421

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1015 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)(7,11)(8,9)(10,12)(13,14,17,22,18,23,34,41)(15,24,27,37,28,40,33,19)(16,29,30,39,31,20,32,21)(25,38,43,35,44,42,26,36);(6,9)(10,11)(13,15,18,28)(14,19,23,37)(16,26,31,43)(17,33,34,27)(20,38,29,42)(21,36,39,35)(22,40,41,24)(25,30,44,32);(5,7,8,12)(6,10,9,11)(13,16)(14,20)(15,25)(17,30)(18,31)(19,35)(21,22)(23,29)(24,42)(26,33)(27,43)(28,44)(32,34)(36,37)(38,40)(39,41);(1,3,4,2)(6,9)(10,11)(13,16)(14,21)(15,26)(17,32)(18,31)(19,36)(20,22)(23,39)(24,42)(25,33)(27,44)(28,43)(29,41)(30,34)(35,37)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285617);