#249053

order	512
nilpotency class	3
isoclinism	isoclinic class of #249019
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1246 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1246 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156249);