#321573

order	512
nilpotency class	3
isoclinism	isoclinic class of #278051
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	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

710 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 198 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)(7,12)(8,17)(9,19)(10,14)(11,16)(13,18)(15,20)(21,22,24,27,25,28,33,36)(23,29,30,34,31,35,32,26);(1,2,3,4)(5,7,9,16,10,15,18,8)(6,11,13,12,14,17,19,20);(2,4)(7,15)(8,16)(11,17)(12,20)(21,23,25,31)(22,26,28,34)(24,32,33,30)(27,35,36,29);(5,8,18,15,10,16,9,7)(6,12,19,11,14,20,13,17)

Polycyclic presentation:

2^9|a^2=f,b^2=egh,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^c=dh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228769);