#343208

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, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

728 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=h,c^2=i,d^2=h,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, 250404);