#844992

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#17086	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dfg,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303956, #72304079, #72304083
maximal normal subgroup of	#72303956, #72304079, #72304083

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 752188);