#190004

order	512
nilpotency class	3
isoclinism	isoclinic class of #188768
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

878 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 234 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,d^b=dh,g^b=gh,g^c=gh

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, 97200);