#353713

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2078 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260909);