#121699

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#121
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=i,e^2=i,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=ei,f^b=fhi,g^b=ghi,d^c=di,e^c=ei,f^c=fhi,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, 28895);