#6457331

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#3054	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364527);