#93178

order	512
nilpotency class	4
isoclinism	isoclinic class of #93165
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×#123	C₂²
prime radical	C₂³×D₄	Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×#123 ▷ C₂⁴ ▷ C₂² ▷ C₁

424 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 44 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:

(1,2,5,9,6,10,4,8)(3,11,13,15,14,16,12,7)(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)(2,7,10,15)(4,12,5,13)(6,14)(8,16)(9,11)(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,g^b=gi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374);