#420062

order	512
nilpotency class	3
isoclinism	isoclinic class of #418504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60456	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1396 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:

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

Polycyclic presentation:

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

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, 327258);