#321876

order	512
nilpotency class	3
isoclinism	isoclinic class of #279051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

745 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:

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

Polycyclic presentation:

2^9|a^2=fh,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^b=dhi,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229072);