#322189

order	512
nilpotency class	3
isoclinism	isoclinic class of #288755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

782 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229385);