#497371

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×#548	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×D₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1159 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)(5,6,10,15,11,16,8,20)(7,12,17,27,19,28,21,35)(9,14,22,30,24,31,23,13)(18,26,33,25,34,36,32,29);(1,2)(3,4)(5,7)(6,12)(8,21)(9,18)(10,17)(11,19)(13,29)(14,26)(15,27)(16,28)(20,35)(22,33)(23,32)(24,34)(25,30)(31,36);(5,8,11,10)(6,13,16,30)(7,17,19,21)(9,23,24,22)(12,25,28,29)(14,15,31,20)(18,33,34,32)(26,35,36,27);(2,4)

Polycyclic presentation:

2^9|a^2=g,c^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, 404567);