#420404

order	512
nilpotency class	3
isoclinism	isoclinic class of #418508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1674 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)(7,19)(8,23)(9,13)(10,17)(11,16)(12,18)(14,27)(15,21)(20,29)(22,30)(24,31)(25,36)(26,33)(28,32)(34,35);(5,7,12,22)(6,13,18,15)(8,24,28,20)(9,10,21,11)(14,29,33,31)(16,30,17,19)(23,34,32,36)(25,27,35,26);(2,4)(5,8,10,26,12,28,11,27)(6,14,16,32,18,33,17,23)(7,20,9,25,22,24,21,35)(13,29,19,34,15,31,30,36);(5,9,10,22,12,21,11,7)(6,15,16,30,18,13,17,19)(8,25,26,24,28,35,27,20)(14,31,32,36,33,29,23,34)

Polycyclic presentation:

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

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, 327600);