#4935915

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2668	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(6,8)(9,10,12,13)(11,15,16,14)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30);(10,13)(14,15)(19,25)(23,26)(27,31)(30,32);(5,6,7,8)(11,16)(14,15)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(9,10)(11,14)(12,13)(15,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=fh,c^2=gh,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72330890, #72330910, #72330923
maximal normal subgroup of	#72330890, #72330910, #72330923
maximal quotient of	#72333280

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4843111);