#212167

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

614 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=gh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^c=dh,f^c=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343666, #72343868, #72343879
maximal normal subgroup of	#72343666, #72343868, #72343879

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119363);