#6447088

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3038	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2420 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

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, 6354284);