#497856

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

600 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)(7,20)(8,25)(9,15)(10,18)(11,27)(12,16)(13,19)(14,23)(17,22)(21,31)(24,26)(28,33)(29,36)(30,34)(32,35);(5,7)(6,14)(8,15)(9,11)(10,30)(12,23)(13,24)(16,26)(17,21)(18,20)(19,34)(22,29)(25,32)(27,36)(28,35)(31,33);(5,8,22,19,12,28,33,16,13,17,25,6,10,27,11,18)(7,21,29,14,23,36,31,20,24,15,32,34,30,35,9,26);(5,9)(6,15)(7,22)(8,26)(10,32)(11,30)(12,29)(13,31)(14,28)(16,36)(17,20)(18,35)(19,21)(23,33)(24,25)(27,34)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405052);