#93108

order	512
nilpotency class	4
isoclinism	isoclinic class of #93099
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator, prime radical	C₂⁵	C₄²
Frattini, cosocle	C₂²×C₂²⋊Q₈	C₂²
LCS	G ▷ C₂⁵ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₂²⋊Q₈ ▷ 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)(3,6,9,13)(5,7,12,15)(10,14,11,16)(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)(2,6,7,14)(4,11,12,9)(5,10)(8,13)(15,16)(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,d^2=hi,e^2=hi,b^a=bc,c^a=cf,e^a=eg,f^a=fh,g^a=gi,c^b=cg,d^b=df,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, 304);