#322619

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 229815);