#93138

order	512
nilpotency class	4
isoclinism	isoclinic class of #93129
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

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

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

normal subgroups only

This group has 406 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,d^2=h,e^2=hi,b^a=bc,c^a=cf,e^a=eghi,f^a=fh,g^a=gi,c^b=cg,d^b=dfhi,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, 334);