#396352

order	512
nilpotency class	3
isoclinism	isoclinic class of #395238
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1401 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)(5,6,10,16)(7,17,20,13)(8,21,23,14)(9,25,11,15)(12,26,28,19)(18,29,33,31)(22,30,24,35)(27,32,34,36);(5,7,9,19,10,20,11,26)(6,12,15,17,16,28,25,13)(8,18,22,32,23,33,24,36)(14,27,30,31,21,34,35,29);(2,4)(5,7,10,20)(6,13,16,17)(8,18,23,33)(9,19,11,26)(12,25,28,15)(14,29,21,31)(22,32,24,36)(27,35,34,30);(5,8,9,22,10,23,11,24)(6,14,15,30,16,21,25,35)(7,18,19,32,20,33,26,36)(12,27,17,31,28,34,13,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303548);