#121629

order	512
nilpotency class	3
isoclinism	isoclinic class of #96924
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=di,e^c=ei,g^c=ghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28825);