#6519807

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2134 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308202, #72308800
maximal normal subgroup of	#72308202, #72308800

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6427003);