#322288

order	512
nilpotency class	3
isoclinism	isoclinic class of #288931
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

762 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

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, 229484);