#95295

order	512
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

500 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=gi,e^2=gh,f^2=gh,b^a=bd,d^a=dg,f^a=fghi,d^b=dh,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239608, #72239616, #72239871
maximal normal subgroup of	#72239608, #72239616, #72239871

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2491);