#495533

order	512
nilpotency class	4
isoclinism	isoclinic class of #481319
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

838 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=f,e^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343243, #72343260, #72343264
maximal normal subgroup of	#72343243, #72343260, #72343264
maximal quotient of	#72343709

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402729);