#186764

order	512
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3548	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93960);