#417700

order	512
nilpotency class	3
isoclinism	isoclinic class of #417700
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2435 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)(3,4)(5,6)(7,17)(8,14)(9,26)(10,15)(11,16)(12,19)(13,24)(18,34)(20,30)(21,29)(22,33)(23,32)(25,31)(27,35)(28,36);(5,7)(6,12)(8,22)(9,20)(10,19)(11,21)(13,28)(14,27)(15,29)(16,30)(17,26)(18,23)(24,35)(25,36)(31,34)(32,33);(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,27)(14,26)(15,31)(16,32)(17,33)(19,22)(20,36)(21,35)(28,29)(30,34);(2,4)(5,8)(6,14)(7,18)(9,24)(10,23)(11,25)(12,28)(13,26)(15,32)(16,31)(17,34)(19,36)(20,22)(21,35)(27,29)(30,33)

Polycyclic presentation:

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

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, 324896);