#426702

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

842 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)(5,6,11,7,10,14,19,17)(8,18,21,28,22,15,26,12)(9,13,25,16,24,29,31,23)(20,32,33,36,34,30,35,27);(5,7,11,14,10,17,19,6)(8,18,21,28,22,15,26,12)(9,23,25,13,24,16,31,29)(20,32,33,36,34,30,35,27);(5,8,19,21,10,22,11,26)(6,12,17,18,14,28,7,15)(9,20,31,33,24,34,25,35)(13,27,23,32,29,36,16,30);(2,4)(5,9)(6,13)(7,16)(8,20)(10,24)(11,25)(12,27)(14,29)(15,30)(17,23)(18,32)(19,31)(21,33)(22,34)(26,35)(28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333898);