#140996

order	512
nilpotency class	4
isoclinism	isoclinic class of #125092
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

344 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,4)(5,6,10,17,11,16,32,33,12,18,19,45,9,15,30,43)(7,13,22,36,21,37,31,57,24,38,53,58,23,35,46,44)(8,25,27,54,28,42,56,65,29,40,47,63,26,14,39,41)(20,48,50,64,49,34,55,59,52,60,62,68,51,61,66,67);(5,7,19,46,9,21,32,22,12,24,10,31,11,23,30,53)(6,13,33,58,16,35,45,57,18,38,43,36,15,37,17,44)(8,20,47,66,26,49,56,50,29,52,27,55,28,51,39,62)(14,34,54,67,25,48,65,68,42,61,63,59,40,60,41,64);(1,2,3,4)(5,8,11,28,12,29,9,26)(6,14,15,40,18,42,16,25)(7,20,23,51,24,52,21,49)(10,27,32,56,19,47,30,39)(13,34,37,60,38,61,35,48)(17,41,43,63,45,65,33,54)(22,50,46,66,53,62,31,55)(36,59,57,68,58,67,44,64)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=ei,d^2=ghi,e^2=i,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,d^b=dh,f^b=fhi,g^b=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48192);