#120808

order	512
nilpotency class	3
isoclinism	isoclinic class of #120808
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
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

618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(2,7,10,22)(3,12)(4,16)(8,23,13,28)(9,21,11,17)(15,27)(18,31)(20,25,29,24);(1,2)(3,13,12,25)(4,17)(5,9)(6,11)(7,18)(8,15,24,27)(10,19)(14,23,26,20)(16,22)(21,31)(28,30,29,32);(1,3,4,14,6,15,18,30)(2,8,7,20,11,25,17,28)(5,12,16,26,19,27,31,32)(9,24,21,23,10,13,22,29)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28004);