#232120

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

569 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=efh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=dei,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343714, #72343868, #72343878
maximal normal subgroup of	#72343714, #72343868, #72343878

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139316);