#249069

order	512
nilpotency class	3
isoclinism	isoclinic class of #249019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#36706	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=hi,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=dhi,f^b=fh,e^c=eh,f^c=fi,f^d=fh

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, 156265);