#495480

order	512
nilpotency class	4
isoclinism	isoclinic class of #33691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

541 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:

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

Polycyclic presentation:

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

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, 402676);