#249158

order	512
nilpotency class	3
isoclinism	isoclinic class of #249139
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	#35150	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1440 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,d^b=dh,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, 156354);