#376941

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	#23438	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1459 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:

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

Polycyclic presentation:

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

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, 284137);