#188067

order	512
nilpotency class	3
isoclinism	isoclinic class of #188062
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2288 conjugacy classes of subgroups; only the 310 normal subgroups are shown.

normal subgroups only

This group has 2288 conjugacy classes of subgroups — too many to draw. Only its 310 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 — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,g^b=gh,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95263);