#322081

order	512
nilpotency class	3
isoclinism	isoclinic class of #288755
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

706 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 194 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=egh,c^2=i,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^c=dh,f^c=fi,g^c=gi

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, 229277);