#7502318

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2217 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409514);