#93124

order	512
nilpotency class	4
isoclinism	isoclinic class of #93119
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×#124	C₂²
prime radical	C₂³×D₄	Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂²×#124 ▷ C₂⁴ ▷ C₂² ▷ C₁

401 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=hi,e^2=h,b^a=bc,c^a=cf,e^a=egi,f^a=fh,g^a=gi,c^b=cg,d^b=dfi,f^b=fi,d^c=dh,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320);