#375880

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#23399	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1664 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:

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

Polycyclic presentation:

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

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, 283076);