#2806657

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2433	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4178 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:

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

Polycyclic presentation:

2^9|a^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293912, C₂×#2806657
maximal normal subgroup of	#72293912, C₂×#2806657
maximal quotient of	#72293958, C₂×#2806657

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2713853);