#425932

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
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	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

952 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)(7,12)(8,21,23,31)(9,14)(10,26)(11,15)(13,24,30,25)(16,29,33,36)(17,20)(18,28)(19,35)(22,27,34,32);(5,7,9,18,11,20,10,19)(6,12,14,28,15,17,26,35)(8,22,24,16,23,34,25,33)(13,29,21,32,30,36,31,27);(5,8,9,25,11,23,10,24)(6,13,14,31,15,30,26,21)(7,16,18,22,20,33,19,34)(12,27,28,36,17,32,35,29);(2,4)(5,6)(7,17)(8,13)(9,14)(10,26)(11,15)(12,20)(16,32)(18,35)(19,28)(21,24)(22,29)(23,30)(25,31)(27,33)(34,36)

Polycyclic presentation:

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

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, 333128);