#140895

order	512
nilpotency class	4
isoclinism	isoclinic class of #125110
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

443 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 102 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)(5,6,10,17,12,18,11,16)(7,19,24,46,25,43,23,47)(8,26,28,14,30,41,29,42)(9,31,33,35,20,44,32,15)(13,34,38,48,39,55,37,22)(21,50,53,65,54,67,52,45)(27,58,60,66,49,40,59,57)(36,62,63,68,64,51,56,61);(5,7,20,48,11,24,32,34,12,25,9,22,10,23,33,55)(6,13,35,19,17,38,31,47,18,39,15,43,16,37,44,46)(8,21,49,62,29,53,59,61,30,54,27,51,28,52,60,68)(14,36,58,45,41,63,57,67,42,64,40,65,26,56,66,50);(1,2,3,4)(5,8,11,29,12,30,10,28)(6,14,17,41,18,42,16,26)(7,21,24,53,25,54,23,52)(9,27,33,60,20,49,32,59)(13,36,38,63,39,64,37,56)(15,40,44,66,35,58,31,57)(19,45,47,67,43,65,46,50)(22,51,55,68,48,62,34,61)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=ei,d^2=gh,e^2=i,f^2=i,g^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,h^b=hi,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, 48091);