#361491

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1435 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)(5,6,10,17,12,16,11,18)(7,13,20,14,8,22,21,28)(9,15,25,30,24,31,26,32)(19,33,34,29,23,27,35,36);(2,4)(5,7,10,21,12,8,11,20)(6,13,18,14,16,22,17,28)(9,19,26,34,24,23,25,35)(15,27,30,29,31,33,32,36);(2,4)(5,8,12,7)(6,14,16,28)(9,23,24,19)(10,20,11,21)(13,18,22,17)(15,29,31,36)(25,34,26,35)(27,30,33,32);(2,4)(5,9,10,26,12,24,11,25)(6,15,18,30,16,31,17,32)(7,19,21,34,8,23,20,35)(13,27,14,29,22,33,28,36)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=i,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 268687);