#121028

order	512
nilpotency class	3
isoclinism	isoclinic class of #10374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#393
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,e^2=hi,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fh,g^b=gh,e^c=ei,f^c=fhi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28224);