#358111

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1766 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,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, 265307);