#360591

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2155 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)(7,19)(8,25)(9,29)(10,17)(11,16)(12,18)(13,22)(14,21)(15,30)(20,34)(23,31)(24,36)(26,35)(27,33)(28,32);(5,7)(6,13)(8,26)(9,21)(10,23)(11,22)(12,24)(14,15)(16,36)(17,19)(18,31)(20,27)(25,29)(28,30)(32,34)(33,35);(5,8,11,21,12,28,10,27)(6,14,17,25,18,33,16,32)(7,20,22,26,24,9,23,30)(13,29,19,35,31,34,36,15);(5,9)(6,15)(7,21)(8,23)(10,26)(11,30)(12,20)(13,14)(16,34)(17,29)(18,35)(19,25)(22,28)(24,27)(31,33)(32,36)

Polycyclic presentation:

2^9|a^2=h,c^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,g^c=gi,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, 267787);