#213518

order	512
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1094 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 1094 conjugacy classes of subgroups — too many to draw. Only its 290 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10,12,15,13,16,11,14)(17,18)(19,23)(20,30,32,24)(21,25)(22,26)(27,41,43,35)(28,36)(29,37)(31,44,46,38)(33,39)(34,40)(42,48,45,47);(1,3)(2,5)(4,8)(6,7)(17,19,22,29)(18,23,26,37)(20,27,32,43)(21,33,34,28)(24,35,30,41)(25,39,40,36)(31,45,46,42)(38,48,44,47);(3,8)(5,7)(17,20,22,32)(18,24,26,30)(19,27,29,43)(21,31,34,46)(23,35,37,41)(25,38,40,44)(28,42,33,45)(36,47,39,48);(2,6)(5,7)(9,11,13,12)(10,14,16,15)(17,21,22,34)(18,25,26,40)(19,28,29,33)(20,31,32,46)(23,36,37,39)(24,38,30,44)(27,42,43,45)(35,47,41,48)

Polycyclic presentation:

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

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, 120714);