#158795

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

4236 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)(6,7)(10,12)(15,19)(18,20)(23,27)(25,28);(9,10)(11,12)(14,17)(15,19)(23,27)(25,28);(5,6,8,7)(13,14)(15,18)(16,17)(19,20)(21,22,24,26)(23,25,27,28);(1,2,3,4)(5,7,8,6)(13,15)(14,18)(16,19)(17,20)(21,23,24,27)(22,25,26,28)

Polycyclic presentation:

2^9|c^2=fgi,d^2=gh,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.

Relationships

maximal subgroup of	#72278433, #72278477, #72278926, …
maximal normal subgroup of	#72278433, #72278477, #72278926, …
maximal quotient of	#72299907

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 65991);