#343149

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
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

816 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=hi,c^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250345);