#191753

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×C₄⋊C₄
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=eh,d^c=dhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98949);