#117796

order	512
nilpotency class	3
isoclinism	isoclinic class of #10144
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

718 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24992);