#393368

order	512
nilpotency class	3
isoclinism	isoclinic class of #32888
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1712 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)(13,14,17,21,18,22,16,20)(15,23,25,19,26,27,24,28);(1,2,3,4)(6,9)(10,11)(13,15)(14,19)(16,25)(17,24)(18,26)(20,27)(21,23)(22,28);(5,7,8,12)(6,10,9,11);(13,15)(14,19)(16,24)(17,25)(18,26)(20,23)(21,27)(22,28)

Polycyclic presentation:

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

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, 300564);