#122217

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₈²	C₂³
socle	C₂²	C₂×#322
prime radical	C₂³	#322
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bf,c^a=ci,e^a=eh,c^b=ch,d^b=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231889, #72231893, #72231894, …
maximal normal subgroup of	#72231889, #72231893, #72231894, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29413);