#321593

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

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

normal subgroups only

This group has 714 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)(7,15)(8,18)(9,20)(10,12)(11,17)(13,19)(14,16)(21,22,25,28,24,26,32,35)(23,29,31,34,30,36,33,27);(1,2,3,4)(5,7,9,17,10,8,16,19)(6,11,14,15,12,13,20,18)(22,26)(27,34)(28,35)(29,36);(2,4)(6,12)(7,8)(14,20)(17,19)(21,23,24,30)(22,27,26,34)(25,33,32,31)(28,36,35,29);(5,8,9,19,10,7,16,17)(6,13,14,18,12,11,20,15)(22,26)(27,34)(28,35)(29,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=eghi,c^2=i,d^2=ei,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, 228789);