#412172

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	#60643	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 319368);