#6545294

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#39334	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2123 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^c=ei,e^d=eh

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, 6452490);