#397515

order	512
nilpotency class	3
isoclinism	isoclinic class of #397514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

1214 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:

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

Polycyclic presentation:

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

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, 304711);