#356053

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2632 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^b=de,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263249);