#195917

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂²×C₂²⋊C₄
prime radical	#34977	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1357 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)(7,14)(8,12)(9,18)(10,13)(11,15)(16,19)(17,20)(21,22,25,28,26,29,24,34)(23,30,31,36,33,35,32,27);(5,7,8,17,10,16,9,15)(6,11,18,19,13,20,12,14)(21,23,26,33)(22,27)(24,32,25,31)(28,30)(29,36)(34,35);(2,4)(5,6)(7,11)(8,18)(9,12)(10,13)(14,15)(16,20)(17,19);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,20,14)(21,24,26,25)(22,28,29,34)(23,31,33,32)(27,35,36,30)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103113);