#185126

order	512
nilpotency class	3
isoclinism	isoclinic class of #160892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂×#3514	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2497 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,e^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=ghi,d^c=dh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92322);