#182834

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1102 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fhi,d^2=i,e^2=h,g^2=hi,b^a=be,e^a=eh,g^a=ghi,c^b=ch,e^b=ei,d^c=di

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, 90030);