#7624397

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#33669	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1209 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,27)(19,23,28,36)(20,24,33,40)(21,34,32,26)(25,41,39,31)(29,35,42,46)(30,45,43,38)(37,48,47,44);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,28)(18,23,27,36)(20,29,33,42)(21,30,32,43)(24,35,40,46)(25,37,39,47)(26,38,34,45)(31,44,41,48);(18,24)(20,31)(21,32)(23,35)(25,26)(27,40)(29,44)(30,43)(33,41)(34,39)(36,46)(37,38)(42,48)(45,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)(19,28)(23,36)(29,42)(30,43)(35,46)(37,47)(38,45)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287781, #72320129, #72320269, …
maximal normal subgroup of	#72287781, #72320129, #72320269, …
maximal quotient of	#72343811

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531593);