#6452999

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
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
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3324 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)(13,14,19,16)(15,23,26,24)(17,22,18,21)(20,27,28,25);(2,4)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,28)(17,27)(18,25)(19,26)(21,24)(22,23);(5,7,9,8)(6,11,12,10)(13,16,19,14)(15,24,26,23)(17,21,18,22)(20,25,28,27);(5,8,9,7)(6,11,12,10)(15,25)(17,18)(20,23)(21,22)(24,28)(26,27);(15,26)(20,28)(23,24)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315268, #72315269, #72315287
maximal normal subgroup of	#72315268, #72315269, #72315287
maximal quotient of	#72341956, #72342155

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6360195);