#195752

order	512
nilpotency class	3
isoclinism	isoclinic class of #18682
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1402 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1402 conjugacy classes of subgroups — too many to draw. Only its 274 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=hi,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dhi,f^a=fi,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102948);