#417662

order	512
nilpotency class	3
isoclinism	isoclinic class of #417661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3027	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2405 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:

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

Polycyclic presentation:

2^9|b^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^c=hi,e^d=ei,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, 324858);