#365778

order	512
nilpotency class	3
isoclinism	isoclinic class of #24987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1332 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:

(2,4)(5,6,11,15)(7,16,14,10)(8,17)(9,23,25,18)(12,26)(13,29,30,24)(19,22)(20,27)(21,28)(31,33)(32,35)(34,36);(5,7,10,6,11,14,16,15)(8,19,26,12,22,17,21,28)(9,18,24,13,25,23,29,30)(20,32,35,27,34,31,33,36);(1,2,3,4)(5,8,11,22)(6,12)(7,17)(9,20,25,34)(10,21,16,26)(13,27)(14,19)(15,28)(18,31)(23,32)(24,33,29,35)(30,36);(5,9,16,29,11,25,10,24)(6,13,7,18,15,30,14,23)(8,20,26,35,22,34,21,33)(12,27,17,31,28,36,19,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272974);