#108234

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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

942 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 942 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 52 points:

(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18)(21,22,25,29,26,30,36,42)(23,27,33,39,32,38,46,50)(24,28,35,41,34,40,47,51)(31,43,45,52,44,37,48,49);(1,2,3,4)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,27)(24,31)(25,33)(26,32)(28,37)(29,39)(30,38)(34,44)(35,45)(36,46)(40,43)(41,49)(42,50)(47,48)(51,52);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24)(22,28)(23,31)(25,35)(26,34)(27,37)(29,41)(30,40)(32,44)(33,45)(36,47)(38,43)(39,49)(42,51)(46,48)(50,52)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,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, 15430);