#420070

order	512
nilpotency class	3
isoclinism	isoclinic class of #418492
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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1368 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)(7,19,23,34)(8,24,15,9)(10,16)(11,17)(12,18)(13,22,33,28)(14,26,30,29)(20,31)(21,25)(27,32)(35,36);(1,2)(3,4)(5,7)(6,13)(8,25)(9,27)(10,28)(11,22)(12,23)(14,21)(15,20)(16,34)(17,19)(18,33)(24,35)(26,36)(29,32)(30,31);(2,4)(5,8)(6,14)(7,20)(9,17)(10,26)(11,29)(12,15)(13,31)(16,24)(18,30)(19,35)(21,33)(22,36)(23,25)(27,34)(28,32);(5,9,10,14,12,24,11,30)(6,15,16,29,18,8,17,26)(7,21,22,27,23,31,28,35)(13,32,19,20,33,36,34,25)

Polycyclic presentation:

2^9|a^2=ef,d^2=e,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^c=fi,h^c=hi,f^d=fi

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, 327266);