#147232

order	512
nilpotency class	4
isoclinism	isoclinic class of #12948
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1864
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dh,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54428);