#6607113

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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2396 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=gi,d^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=eh,d^c=dhi,e^c=ei,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, 6514309);