#93373

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#99
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂×#376	C₂²
socle	C₂³	#342
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂×#376 ▷ C₂⁴ ▷ C₂ ▷ C₁

414 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=i,e^2=h,b^a=bc,c^a=cf,e^a=efg,c^b=cg,g^b=gi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 569);