#298252

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#29794	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=f,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi

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, 205448);