#206700

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

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 ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

590 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)(11,17)(12,20)(14,21)(16,22)(18,23)(19,24)(25,26,29,34,30,35,39,33)(27,31,28,37,36,32,38,40);(2,5)(6,8)(9,10)(11,18)(12,16)(13,15)(14,19)(17,23)(20,22)(21,24)(25,27,29,28,30,36,39,38)(26,31,34,37,35,32,33,40);(9,11,12,17,13,19,20,24)(10,14,16,21,15,18,22,23)(25,28,29,36,30,38,39,27)(26,32,33,37,35,31,34,40);(1,3,4,7)(2,6,5,8)(10,15)(11,19)(16,22)(17,24)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343751, #72343872, #72343877
maximal normal subgroup of	#72343751, #72343872, #72343877

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113896);