#375421

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
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

1008 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 242 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=eh,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,g^c=gi

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, 282617);