#6607122

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2455 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)(5,6)(7,10)(8,11)(9,12)(14,20)(15,22)(17,18)(21,28)(23,24)(26,27);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,14,18,15,19,22,17,20)(16,24,27,28,25,21,26,23);(7,12)(9,10)(13,15,18,22,19,20,17,14)(16,23,27,24,25,28,26,21);(5,7)(6,10)(8,12)(9,11)(13,16,19,25)(14,21,22,24)(15,23,20,28)(17,26,18,27);(7,12)(9,10)(14,22)(15,20)(16,25)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263948, #72263976, #72264026
maximal normal subgroup of	#72263948, #72263976, #72264026
maximal quotient of	#72264342

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6514318);