#190101

order	512
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 964 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)(3,6)(4,11,12,7)(5,8)(9,15,16,13)(10,14)(17,18)(19,27,30,23)(20,31)(21,25)(22,26)(24,33)(28,35,42,39)(29,40,43,36)(32,44)(34,38)(37,45)(41,46,48,47);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,35)(25,36)(26,27)(31,39)(32,41)(33,42)(34,43)(37,46)(38,40)(44,47)(45,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,20)(18,24,26,31)(19,28)(21,32)(22,33)(23,35,27,39)(25,37,38,44)(29,41)(30,42)(34,45)(36,46,40,47)(43,48);(17,21,22,34)(18,25,26,38)(19,29,30,43)(20,32,33,45)(23,36,27,40)(24,37,31,44)(28,41,42,48)(35,46,39,47)

Polycyclic presentation:

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

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, 97297);