#182673

order	512
nilpotency class	3
isoclinism	isoclinic class of #17547
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	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,e^2=h,b^a=be,e^a=eh,g^a=ghi,e^b=ei,d^c=dhi

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89869);