#318424

order	512
nilpotency class	3
isoclinism	isoclinic class of #262883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

580 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eghi,c^2=i,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,d^c=dh,f^c=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, 225620);