#186022

order	512
nilpotency class	3
isoclinism	isoclinic class of #18162
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×#3493	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2418 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)(5,6,9,11,10,7,8,12)(13,14)(15,16,23,19)(17,25)(18,21)(20,27)(22,26,28,24);(2,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,21)(17,22)(18,23)(20,26)(24,25)(27,28);(13,16,18,19)(14,15,21,23)(17,26,27,24)(20,28,25,22);(5,8,10,9)(6,11,7,12)(13,17,18,27)(14,20,21,25)(15,22,23,28)(16,24,19,26)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,d^c=dh,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, 93218);