#6445130

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	#35195	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2077 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,d^c=di

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, 6352326);