#6470018

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4459 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,25)(15,21,16,22)(17,24,18,23)(20,27,28,26);(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,19)(17,26)(18,27)(21,23)(22,24)(25,28);(2,4)(7,11)(9,12)(13,15,19,16)(14,21,25,22)(17,27,18,26)(20,24,28,23);(7,11)(9,12)(13,16,19,15)(14,22,25,21)(17,26,18,27)(20,23,28,24);(6,10)(9,12)(13,17,19,18)(14,23,25,24)(15,26,16,27)(20,21,28,22)

Polycyclic presentation:

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

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, 6377214);