#6606991

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6514187);