#107149

order	512
nilpotency class	3
isoclinism	isoclinic class of #8453
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1786 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1786 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 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)(13,14,18,23,19,24,17,22)(15,25,29,20,28,35,27,34)(16,21,30,36,31,37,32,38)(26,40,41,44,42,39,43,33);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,26)(17,29)(18,27)(19,28)(21,33)(22,35)(23,25)(24,34)(30,43)(31,42)(32,41)(36,39)(37,44)(38,40);(13,16,19,31)(14,21,24,37)(15,26,28,42)(17,32,18,30)(20,33,34,44)(22,38,23,36)(25,39,35,40)(27,43,29,41)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=ei,f^b=fhi,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, 14345);