#504840

order	512
nilpotency class	4
isoclinism	isoclinic class of #504384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13316	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1753 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

603 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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=efh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=gh,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412036);