#393321

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2036 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=fi,b^a=be,c^a=cf,d^a=dg,g^a=gi,g^b=gi,e^d=ei,h^d=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300517);