#6459794

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35513	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3207 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gh,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6366990);