#120232

order	512
nilpotency class	3
isoclinism	isoclinic class of #120231
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

531 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)(9,13,16,15)(10,14)(17,18,22,26,23,28,21,27)(19,29,32,41,33,40,31,24)(20,34,36,44,38,43,37,25)(30,39,35,42,46,48,47,45);(1,3)(2,6)(4,9)(5,10)(7,15)(8,14)(11,13)(12,16)(17,19,21,31,23,33,22,32)(18,24,26,29,28,41,27,40)(20,35,36,46,38,47,37,30)(25,42,43,39,44,45,34,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,20,21,36,23,38,22,37)(18,25,26,43,28,44,27,34)(19,30,31,35,33,46,32,47)(24,39,29,45,41,48,40,42)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^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,e^b=ei,f^b=fh,f^c=fh,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27428);