#212782

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
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	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

623 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)(10,14)(11,17)(12,20)(15,18)(16,22)(19,23)(25,26,29,33,30,34,38,32)(27,35,28,31,36,39,37,40);(2,5)(6,8)(9,10,12,14,13,16,20,22)(11,18,17,24,19,15,23,21)(25,27,29,28,30,36,38,37)(26,31,32,35,34,40,33,39);(1,3,4,7)(2,6,5,8)(9,11,12,17,13,19,20,23)(10,15,14,21,16,18,22,24)(25,27,29,28,30,36,38,37)(26,31,32,35,34,40,33,39);(11,19)(15,18)(17,23)(21,24)(25,28,29,36,30,37,38,27)(26,31,33,39,34,40,32,35)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=g,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=chi,d^b=di,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343692, #72343872, #72343877
maximal normal subgroup of	#72343692, #72343872, #72343877

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119978);