#94103

order	512
nilpotency class	5
isoclinism	isoclinic class of #1414
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#354
commutator	C₄²	C₄×C₈
Frattini, cosocle	C₂×#382	C₂²
prime radical	C₂⁴	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#382 ◁ G
LECS	G ▷ C₂×#382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

322 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 2 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=dfhi,d^c=dhi,e^d=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1299);