#6552532

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#38089	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1905 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,g^c=gi,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, 6459728);