#7065155

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	C₂×#3306	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1938 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=hi,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6972351);