#188018

order	512
nilpotency class	3
isoclinism	isoclinic class of #187944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3485	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

3440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260842
maximal normal subgroup of	#72260842
maximal quotient of	#72261056

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95214);