#484050

order	512
nilpotency class	4
isoclinism	isoclinic class of #484039
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33675	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

427 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dhi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216790, #72216957, #72216968, …
maximal normal subgroup of	#72216790, #72216957, #72216968, …
maximal quotient of	#72343991

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391246);