#283124

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

695 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 190320);