#497322

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	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#550 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,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,d^c=di,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, 404518);