#249296

order	512
nilpotency class	3
isoclinism	isoclinic class of #249259
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1232 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 52 points:

(5,6)(7,11)(8,12)(9,17)(10,13)(14,20)(15,18)(16,19)(21,22,26,30)(23,27,32,41)(24,34,36,44)(25,37,39,29)(28,38,43,35)(31,45,46,52)(33,49,48,42)(40,50,51,47);(1,2)(3,4)(5,7)(6,11)(8,14)(9,10)(12,18)(13,17)(15,16)(19,20)(21,23,26,32)(22,27,30,41)(24,31,36,46)(25,33,39,48)(28,40,43,51)(29,42,37,49)(34,45,44,52)(35,47,38,50);(2,4)(5,8)(6,12)(7,14)(9,15)(10,16)(11,18)(13,19)(17,20)(21,24,26,36)(22,28,30,43)(23,31,32,46)(25,38,39,35)(27,40,41,51)(29,44,37,34)(33,50,48,47)(42,52,49,45);(5,9,10,7)(6,11,13,17)(8,15,16,14)(12,18,19,20)(23,32)(27,41)(31,46)(33,48)(40,51)(42,49)(45,52)(47,50)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,d^b=dhi,f^b=fh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156492);