#412904

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

880 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=e,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320100);