#188682

order	512
nilpotency class	3
isoclinism	isoclinic class of #161262
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

982 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)(5,6,9,11,10,7,8,12)(13,14)(15,25)(16,20)(17,24)(18,23)(19,22)(21,27)(26,28);(1,2,3,4)(13,15,18,16)(14,20,23,25)(17,27,19,26)(21,24,28,22);(5,7)(6,10)(8,12)(9,11)(13,16,18,15)(14,21,23,28)(17,27,19,26)(20,24,25,22);(5,8,10,9)(6,11,7,12)(13,17,18,19)(14,22,23,24)(15,26,16,27)(20,28,25,21)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=h,d^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95878);