#417680

order	512
nilpotency class	3
isoclinism	isoclinic class of #417534
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
prime radical	C₂×#3019	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2474 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)(5,6,11,16)(7,17,20,28)(8,13,24,31)(9,15,10,14)(12,21,27,19)(18,33,34,29)(22,36,35,26)(23,32,25,30);(5,7)(6,12)(8,22)(9,21)(10,19)(11,20)(13,29)(14,28)(15,17)(16,27)(18,23)(24,35)(25,34)(26,30)(31,33)(32,36);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,30)(15,32)(16,31)(17,29)(19,22)(20,34)(21,35)(27,36)(28,33);(2,4)(5,8)(6,13)(7,18)(9,25)(10,23)(11,24)(12,26)(14,32)(15,30)(16,31)(17,33)(19,35)(20,34)(21,22)(27,36)(28,29)

Polycyclic presentation:

2^9|a^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,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, 324876);