#497370

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#37680	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#548 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1179 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404566);