#6591661

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	#39462	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2046 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:

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

Polycyclic presentation:

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

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, 6498857);