#183854

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator, socle	C₂³	C₂²×C₄²
prime radical	C₂²×D₄	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

768 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)(5,6,9,14,10,12,19,15)(7,11,16,8,13,17,20,18)(21,22)(23,27)(24,28)(25,30)(26,29)(31,34)(32,35)(33,36);(2,4)(5,7)(6,11)(8,14)(9,16)(10,13)(12,17)(15,18)(19,20)(21,23,25,24,26,31,33,32)(22,27,30,28,29,34,36,35);(6,12)(7,13)(14,15)(16,20)(21,24,25,31,26,32,33,23)(22,28,30,34,29,35,36,27);(5,8,9,17,10,18,19,11)(6,13,14,20,12,7,15,16)(22,29)(27,34)(28,35)(30,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,c^b=chi,d^b=dh

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, 91050);