#93041

order	512
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#327
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×#431	C₂²
socle	C₂²	#1294
prime radical	C₂⁴	#107
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×#431 ▷ C₂²×C₄ ▷ C₂² ▷ C₁

226 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=i,e^2=i,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,e^c=ehi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237);