#497853

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
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

606 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)(6,12)(7,17)(9,25)(10,14)(13,27)(15,16)(18,26)(19,36)(20,33)(21,31)(22,23)(28,32)(29,35)(30,34);(5,6,11,16)(7,13,20,30)(8,21,24,10)(9,15,25,12)(14,23,31,22)(17,18,33,35)(19,29,32,26)(27,28,34,36);(5,7,14,28,9,19,21,33,11,20,31,36,25,32,10,17)(6,13,23,34,15,29,24,35,16,30,22,27,12,26,8,18);(5,8)(6,14)(7,18)(9,23)(10,12)(11,24)(13,28)(15,21)(16,31)(17,26)(19,34)(20,35)(22,25)(27,32)(29,33)(30,36)

Polycyclic presentation:

2^9|a^2=g,b^2=i,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,d^b=de,e^b=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405049);