#363974

order	512
nilpotency class	3
isoclinism	isoclinic class of #23998
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2604 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)(7,11,16,27)(8,18)(9,21)(10,13)(12,19)(14,24)(15,32,33,36)(17,30,22,26)(20,31)(23,29)(25,35,28,34);(2,4)(5,7)(6,11,13,27)(8,15)(9,22)(10,16)(12,28,29,25)(14,30,21,26)(17,24)(18,32,31,36)(19,35)(20,33)(23,34);(5,8,10,20)(6,12,13,29)(7,15,16,33)(9,23,24,19)(11,25,27,28)(14,31,21,18)(17,35,22,34)(26,36,30,32);(2,4)(5,7)(6,11,13,27)(8,15)(9,17)(10,16)(12,25,29,28)(14,26,21,30)(18,32,31,36)(19,34)(20,33)(22,24)(23,35)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,f^c=fi,e^d=ei,h^d=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, 271170);