#4941680

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3994 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)(5,6,8,9)(7,10,11,12)(13,14,16,17)(15,19,20,18)(21,22,25,27)(23,24,28,26);(6,9)(10,12)(14,17)(18,19)(21,23)(22,24)(25,28)(26,27);(7,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,23,25,28)(22,26,27,24);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,20)(17,19)(21,23)(22,24)(25,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,26)(27,28)

Polycyclic presentation:

2^9|a^2=fgi,c^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	#72326451, #72326498, #72326523
maximal normal subgroup of	#72326451, #72326498, #72326523
maximal quotient of	#72328671

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4848876);