#7623518

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623305
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	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1305 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)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,23,26)(19,24,22,37)(20,30,32,46)(21,33,36,47)(25,40,41,31)(27,29,43,35)(28,45,38,44)(34,39,48,42);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,23,22)(18,24,26,37)(20,28,32,38)(21,29,36,35)(25,39,41,42)(27,33,43,47)(30,45,46,44)(31,34,40,48);(3,10)(6,9)(11,16)(13,15)(17,20)(18,25)(19,28)(21,34)(22,38)(23,32)(24,39)(26,41)(27,44)(29,40)(30,33)(31,35)(36,48)(37,42)(43,45)(46,47);(2,7)(6,9)(8,14)(13,15)(17,19,23,22)(18,24,26,37)(20,28,32,38)(21,29,36,35)(25,39,41,42)(27,33,43,47)(30,45,46,44)(31,34,40,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(18,26)(24,37)(25,41)(27,43)(30,46)(33,47)(39,42)(44,45)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287695, #72287701, #72287704, …
maximal normal subgroup of	#72287695, #72287701, #72287704, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530714);