#6617643

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3283 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)(13,14,19,22,18,23,17,21)(15,24,25,20,27,28,26,16);(6,10)(8,11)(13,15)(14,20)(16,23)(17,25)(18,27)(19,26)(21,28)(22,24);(2,4)(5,6)(7,8)(9,10)(11,12)(13,16,18,20)(14,15,23,27)(17,28,19,24)(21,26,22,25);(5,7,9,12)(6,11,10,8)(13,17,18,19)(14,21,23,22)(15,25,27,26)(16,24,20,28);(5,8,9,11)(6,7,10,12)(13,17,18,19)(14,22,23,21)(15,25,27,26)(16,28,20,24)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305352, #72305363, #72305374
maximal normal subgroup of	#72305352, #72305363, #72305374
maximal quotient of	#72305768, #72305771, #72305779, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6524839);