#343364

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
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

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

normal subgroups only

This group has 722 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250560);