#321827

order	512
nilpotency class	3
isoclinism	isoclinic class of #278987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

707 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 707 conjugacy classes of subgroups — too many to draw. Only its 162 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)(7,12)(8,17)(9,19)(10,14)(11,16)(13,18)(15,20)(21,22,26,32,27,33,34,31)(23,30,35,28,29,24,25,36);(1,2,3,4)(5,7,9,16,10,15,18,8)(6,11,13,12,14,17,19,20)(21,23,34,25,27,29,26,35)(22,28,32,24,33,36,31,30);(2,4)(7,15)(8,16)(11,17)(12,20)(21,24,27,30)(22,29,33,23)(25,32,35,31)(26,28,34,36);(5,8,18,15,10,16,9,7)(6,12,19,11,14,20,13,17)(21,25,34,29,27,35,26,23)(22,30,32,28,33,24,31,36)

Polycyclic presentation:

2^9|a^2=f,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,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, 229023);