#412617

order	512
nilpotency class	3
isoclinism	isoclinic class of #394463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#397
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 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:

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

Polycyclic presentation:

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

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, 319813);