#277960

order	512
nilpotency class	3
isoclinism	isoclinic class of #277931
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3100	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,8,11,7,13,6,12)(3,4,5,15,10,16,14,9)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(1,3,6,14,7,10,8,5)(2,9,11,4,13,15,12,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,29)(31,32);(1,4)(2,10)(3,13)(5,11)(6,15)(7,16)(8,9)(12,14)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28);(1,5,6,3,7,14,8,10)(2,4,11,15,13,16,12,9)(18,23)(22,25)(24,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185156);