#429982

order	512
nilpotency class	3
isoclinism	isoclinic class of #419450
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1544 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)(7,11,16,26)(8,12)(9,21)(10,13)(14,24)(15,33,34,25)(17,31,22,27)(18,28,35,36)(19,32)(20,29)(23,30);(5,7)(6,11,13,26)(8,18)(9,22)(10,16)(12,28,29,36)(14,31,21,27)(15,19)(17,24)(20,35)(23,34)(25,32,33,30);(5,8,10,20)(6,12,13,29)(7,15,16,34)(9,23,24,19)(11,25,26,33)(14,32,21,30)(17,35,22,18)(27,36,31,28);(2,4)(5,6)(7,11,16,26)(8,12)(9,14)(10,13)(15,25,34,33)(17,27,22,31)(18,28,35,36)(19,30)(20,29)(21,24)(23,32)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=f,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, 337178);