#6577583

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#23929	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2092 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6484779);