#6393839

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3543	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3528 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)(6,9)(7,11)(13,14)(15,22)(16,17)(18,21)(19,26)(20,25)(23,24)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,23)(17,27)(18,25)(19,24)(21,22)(26,28);(2,4)(13,16,17,14,18,26,19,21)(15,23,27,20,25,28,24,22);(6,9)(10,12)(13,17,18,19)(14,16,21,26)(15,24,25,27)(20,28,22,23);(5,7,8,11)(6,10,9,12)(14,21)(15,25)(16,26)(24,27)

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6301035);