#93842

order	512
nilpotency class	5
isoclinism	isoclinic class of #93790
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1344
commutator	C₂×C₄×C₈	C₂×C₄
Frattini, cosocle	#1984	C₂²
prime radical	C₂³	#327
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1984 ◁ G
LECS	G ▷ #1984 ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁

128 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 128

A minimal faithful permutation representation — 2 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=gh,c^2=fh,d^2=h,e^2=h,f^2=hi,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,d^b=defhi,e^b=eh,f^b=fhi,d^c=dgh,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1038);