#6607041

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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

2380 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)(15,24,27,22)(16,21,26,25)(18,23,19,28);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,26,20,27,19,16)(14,21,28,22,17,25,23,24);(7,12)(9,10)(13,16,18,15,20,26,19,27)(14,22,28,25,17,24,23,21);(5,7)(6,10)(8,12)(9,11)(13,17,20,14)(15,25,27,21)(16,22,26,24)(18,23,19,28);(7,12)(9,10)(13,18,20,19)(14,23,17,28)(15,26,27,16)(21,24,25,22)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=gi,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,e^b=ehi,d^c=dh,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6514237);