#497376

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	#37683	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#550 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1200 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)(7,12,18,28,21,29,19,26)(9,15,22,31,24,32,23,13)(20,27,34,25,35,36,33,30);(1,2)(3,4)(5,7,11,21)(6,12,17,29)(8,19,10,18)(9,20,24,35)(13,30,31,25)(14,26,16,28)(15,27,32,36)(22,34,23,33);(5,8,11,10)(6,13,17,31)(7,18,21,19)(9,23,24,22)(12,25,29,30)(14,15,16,32)(20,34,35,33)(26,36,28,27);(2,4)(5,8,11,10)(6,14,17,16)(7,19,21,18)(9,22,24,23)(12,26,29,28)(13,15,31,32)(20,34,35,33)(25,36,30,27)

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,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, 404572);