#6345381

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2920 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,d^a=dh,e^a=ei,f^a=fh,f^b=fh,d^c=dhi,e^c=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306926, #72306932, #72307028, …
maximal normal subgroup of	#72306926, #72306932, #72307028, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252577);