#6583308

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
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1770 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,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	#72263950, #72263954, #72263970
maximal normal subgroup of	#72263950, #72263954, #72263970
maximal quotient of	#72307867

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490504);