#115325

order	512
nilpotency class	3
isoclinism	isoclinic class of #115318
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=dh,f^b=fh,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22521);