#455693

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362889);