#368129

order	512
nilpotency class	3
isoclinism	isoclinic class of #23882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 275325);