#248929

order	512
nilpotency class	3
isoclinism	isoclinic class of #248899
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	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1472 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1472 conjugacy classes of subgroups — too many to draw. Only its 188 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=dh,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, 156125);