#122415

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
socle	C₂²	#1459
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

301 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=gh,d^2=hi,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,d^b=dhi,e^b=efhi,e^d=eh

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, 29611);