#6577934

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2029 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)(7,11)(8,12)(9,10)(13,14,18,20)(15,21,26,27)(16,22,19,24)(17,23,28,25);(6,10)(11,12)(14,20)(15,25)(16,19)(17,27)(21,28)(23,26);(2,4)(13,15,18,26)(14,21,20,27)(16,25,19,23)(17,24,28,22);(5,7,9,8)(6,11,10,12)(13,16)(14,22)(17,28)(18,19)(20,24)(23,25);(5,8,9,7)(6,12,10,11)(13,17,18,28)(14,23,20,25)(15,24,26,22)(16,27,19,21)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263909, #72263950, #72263990
maximal normal subgroup of	#72263909, #72263950, #72263990
maximal quotient of	#72265171, #72308242

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485130);