#348238

order	512
nilpotency class	3
isoclinism	isoclinic class of #346398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh,f^c=fh,g^c=gi,f^d=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, 255434);