#495542

order	512
nilpotency class	4
isoclinism	isoclinic class of #33688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

825 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:

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

Polycyclic presentation:

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

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, 402738);