#115336

order	512
nilpotency class	3
isoclinism	isoclinic class of #9946
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258158, #72258163
maximal normal subgroup of	#72258158, #72258163

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22532);