#6432851

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432843
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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 ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4387 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)(15,23)(16,21)(17,20)(18,22)(19,25)(24,27)(26,28);(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,25)(17,27)(18,26)(20,23)(21,28)(22,24);(2,4)(7,11)(9,12)(13,14)(15,24)(16,20)(17,21)(18,22)(19,26)(23,27)(25,28);(6,10)(9,12)(13,16,18,17)(14,20,22,21)(15,25,26,27)(19,23,24,28);(13,16,18,17)(14,21,22,20)(15,25,26,27)(19,28,24,23)

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,e^c=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, 6340047);