#6584771

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1991 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263921, #72263954, #72263962
maximal normal subgroup of	#72263921, #72263954, #72263962
maximal quotient of	#72265008

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491967);