#7021584

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2325 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)(7,9)(8,10)(11,12)(13,14,19,23,20,24,18,15)(16,17,22,25,26,21,28,27);(5,6)(7,11)(8,10)(9,12)(13,15)(14,18)(16,25)(17,22)(19,24)(20,23)(21,28)(26,27);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,16,20,26)(14,21,24,17)(15,25,23,27)(18,28,19,22);(7,12)(9,11)(13,17,19,25,20,21,18,27)(14,22,23,26,24,28,15,16);(6,10)(9,11)(13,18,20,19)(14,23,24,15)(16,22,26,28)(17,27,21,25)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6928780);