#382427

order	512
nilpotency class	3
isoclinism	isoclinic class of #184805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×Q₈ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3574 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

normal subgroups only

This group has 3574 conjugacy classes of subgroups — too many to draw. Only its 362 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:

(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,26)(17,21)(18,22)(20,28)(23,37)(24,32)(25,33)(27,40)(29,35)(30,36)(31,39)(34,42)(38,44)(41,43);(1,2)(3,4)(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);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,20,22,26)(15,23)(17,27)(18,28)(19,31,33,37)(21,34,36,40)(24,38)(25,39)(29,41)(30,42)(32,43,35,44);(7,11)(10,12)(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|b^2=i,c^2=h,d^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,d^b=di,d^c=de,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289623);