#193594

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1173 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:

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

Polycyclic presentation:

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

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, 100790);