#180868

order	512
nilpotency class	2
isoclinism	isoclinic class of #180864
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=i,b^a=be,g^a=gh,c^b=ci,f^b=fh,d^c=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303255, #72303270, #72303288, …
maximal normal subgroup of	#72303255, #72303270, #72303288, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88064);