#108237

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	C₂×C₄	#379
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

900 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,26,23,28,22,27)(19,24,31,38,30,37,32,39)(20,25,34,41,33,40,35,42)(29,43,45,48,44,36,46,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,32,23,30,21,31)(18,24,27,39,28,37,26,38)(20,29,35,46,33,44,34,45)(25,36,42,48,40,43,41,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,36)(26,41)(27,42)(28,40)(30,44)(31,45)(32,46)(37,43)(38,47)(39,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,f^2=i,g^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, 15433);