#388792

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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

1061 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=eh,e^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295988);