#116504

order	512
nilpotency class	3
isoclinism	isoclinic class of #116502
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

710 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 124 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ehi,e^c=eh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23700);