#479969

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	#97
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

519 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=g,f^2=h,h^2=i,b^a=bh,d^a=di,f^a=fi,c^b=cg,e^b=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286699, #72286718, #72286780, …
maximal normal subgroup of	#72286699, #72286718, #72286780, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387165);