#492597

order	512
nilpotency class	4
isoclinism	isoclinic class of #492504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33838	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,6)(3,4)(7,8)(9,10,14,17)(11,15,22,28)(12,16,25,31)(13,26,24,19)(18,33,32,23)(20,27,36,38)(21,37,35,30)(29,40,39,34);(1,2,5,7)(3,6,4,8)(9,11,14,22)(10,15,17,28)(12,20,25,36)(13,21,24,35)(16,27,31,38)(18,29,32,39)(19,30,26,37)(23,34,33,40);(10,16)(12,23)(13,24)(15,27)(17,31)(18,19)(20,34)(21,35)(25,33)(26,32)(28,38)(29,30)(36,40)(37,39);(1,3)(4,5)(6,8)(10,17)(15,28)(16,31)(18,32)(19,26)(27,38)(29,39)(30,37)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309684, #72309977
maximal normal subgroup of	#72309684, #72309977

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399793);