#186024

order	512
nilpotency class	3
isoclinism	isoclinic class of #18252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3558	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,g^2=i,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,g^c=gh

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, 93220);