#576935

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂×#2424	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333920, #72333921, #72333950
maximal normal subgroup of	#72333920, #72333921, #72333950

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 484131);