#6606619

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#39483	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1853 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513815);