#7198642

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021351
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³	C₂×H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1547 conjugacy classes of subgroups; only the 508 normal subgroups are shown.

normal subgroups only

This group has 1547 conjugacy classes of subgroups — too many to draw. Only its 508 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 — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=ghi,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7105838);