#392878

order	512
nilpotency class	3
isoclinism	isoclinic class of #32878
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1790 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)(7,8)(9,10,12,11)(13,14)(15,22)(16,21)(17,25)(18,20)(19,23)(24,28)(26,27);(9,11,12,10)(13,15)(14,19)(16,23)(17,26)(18,24)(20,27)(21,28)(22,25);(1,2,3,4)(14,20)(19,27)(21,25)(22,28);(6,8)(13,16,18,17)(14,21,20,25)(15,23,24,26)(19,28,27,22)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300074);