#496675

order	512
nilpotency class	4
isoclinism	isoclinic class of #33758
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁵	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

770 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 250 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)(7,11)(8,10)(9,12)(13,14)(15,20)(16,26)(17,31)(18,33)(19,23)(21,29)(22,27)(24,37)(25,36)(28,39)(30,43)(32,44)(34,38)(35,42)(40,41);(5,7)(6,9)(8,12)(10,11)(13,15,19,18)(14,20,23,33)(16,24,28,30)(17,25,32,27)(21,34,35,40)(22,31,36,44)(26,37,39,43)(29,38,42,41);(7,12)(9,11)(13,16,27,41,18,30,32,42,19,28,25,38,15,24,17,29)(14,21,31,37,20,34,36,39,23,35,44,43,33,40,22,26);(6,10)(9,11)(16,28)(21,35)(24,30)(26,39)(29,42)(34,40)(37,43)(38,41)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fh,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di

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, 403871);