#497084

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

496 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)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22,26,31,27,23,32,33,25,30,34,36,24,29,28,35);(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,23,26,33,27,30,32,36,25,29,34,35,24,22,28,31);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,24,25,27)(22,29,30,23)(26,28,34,32)(31,35,36,33);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(21,24,25,27)(22,29,30,23)(26,28,34,32)(31,35,36,33)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ef,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404280);