#358344

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,11,14,10,15,9,16)(7,17,21,27,19,28,20,12)(8,13,24,30,25,31,23,32)(18,29,22,35,33,36,34,26);(1,2)(3,4)(5,7,11,21,10,19,9,20)(6,12,16,28,15,27,14,17)(8,22,23,18,25,34,24,33)(13,29,30,35,31,36,32,26);(2,4)(5,8,9,23,10,25,11,24)(6,13,14,30,15,31,16,32)(7,18,20,34,19,33,21,22)(12,26,17,29,27,35,28,36);(7,19)(12,27)(17,28)(18,33)(20,21)(22,34)(26,35)(29,36)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=di,f^c=fi,g^c=gi

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, 265540);