#6597105

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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	#37659	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2224 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504301);