#204881

order	512
nilpotency class	3
isoclinism	isoclinic class of #161894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1546 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fi,d^2=hi,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dh,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112077);