#213003

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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

770 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)(5,6,10,13)(7,14,12,9)(8,11,16,18)(15,20,19,17)(21,22,27,30,26,29,25,34)(23,31,33,35,32,36,24,28);(5,7,9,6,10,12,14,13)(8,15,17,11,16,19,20,18)(21,23,25,24,26,32,27,33)(22,28,30,31,29,35,34,36);(1,2,3,4)(5,8,10,16)(6,11,13,18)(7,15,12,19)(9,17,14,20)(22,29)(23,32)(24,33)(30,34);(8,16)(11,18)(15,19)(17,20)(21,24,25,32,26,33,27,23)(22,28,34,36,29,35,30,31)

Polycyclic presentation:

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

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, 120199);