#156775

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

3248 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:

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

Polycyclic presentation:

2^9|a^2=eg,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 63971);