#108233

order	512
nilpotency class	3
isoclinism	isoclinic class of #8534
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

(5,6)(7,11,12,9)(8,10)(13,14,17,21,18,22,28,34)(15,19,25,31,24,30,38,42)(16,20,27,33,26,32,39,43)(23,35,37,44,36,29,40,41);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,23)(17,25)(18,24)(20,29)(21,31)(22,30)(26,36)(27,37)(28,38)(32,35)(33,41)(34,42)(39,40)(43,44);(13,16)(14,20)(15,23)(17,27)(18,26)(19,29)(21,33)(22,32)(24,36)(25,37)(28,39)(30,35)(31,41)(34,43)(38,40)(42,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^b=ei,f^b=fh,d^c=di

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, 15429);