#428002

order	512
nilpotency class	3
isoclinism	isoclinic class of #417690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1280 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335198);