#351741

order	512
nilpotency class	3
isoclinism	isoclinic class of #347668
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

682 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=f,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,d^c=dh,g^d=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, 258937);