#374094

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(5,6,7,8)(9,10)(11,19)(12,26)(13,31)(14,21)(15,20)(16,22)(17,24)(18,29)(23,37)(25,36)(27,40)(28,35)(30,34)(32,39)(33,38);(1,2)(3,4)(9,11)(10,17)(12,23)(13,14)(15,24)(16,25)(18,32)(19,21)(20,36)(22,31)(26,33)(27,28)(29,38)(30,39)(34,40)(35,37);(2,4)(6,8)(9,12,15,29,16,30,14,28)(10,18,21,26,22,35,20,34)(11,23,24,38,25,39,13,27)(17,32,19,33,31,37,36,40);(9,13,16,24)(10,19,22,36)(11,14,25,15)(12,27,30,38)(17,21,31,20)(18,33,35,40)(23,28,39,29)(26,37,34,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281290);