#6443704

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	#35563	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2450 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:

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

Polycyclic presentation:

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

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, 6350900);