#392212

order	512
nilpotency class	3
isoclinism	isoclinic class of #184786
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×Q₈ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3458 conjugacy classes of subgroups; only the 442 normal subgroups are shown.

normal subgroups only

This group has 3458 conjugacy classes of subgroups — too many to draw. Only its 442 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)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,26,28,20)(17,21)(18,22)(23,37,39,31)(24,32)(25,33)(27,40,42,34)(29,35)(30,36)(38,44,41,43);(6,9)(10,12)(13,15,18,25)(14,19,22,33)(16,23,28,39)(17,29,30,24)(20,31,26,37)(21,35,36,32)(27,41,42,38)(34,44,40,43);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,23)(17,27)(18,28)(19,31)(21,34)(22,26)(24,38)(25,39)(29,41)(30,42)(32,43)(33,37)(35,44)(36,40);(2,4)(13,17,18,30)(14,21,22,36)(15,24,25,29)(16,27,28,42)(19,32,33,35)(20,34,26,40)(23,38,39,41)(31,43,37,44)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,d^b=di,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299408);