#2328212

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3328 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)(7,12)(8,11)(9,10)(13,14,16,19)(15,18,20,17)(21,22,27,28)(23,25,26,24);(5,7,10,8)(6,11,9,12)(13,15)(14,17)(16,20)(18,19)(21,23,27,26)(22,24,28,25);(6,9)(11,12)(15,20)(17,18)(21,24,27,25)(22,23,28,26);(1,2,3,4)(5,8,10,7)(6,12,9,11)(13,14)(15,18)(16,19)(17,20)(21,25,27,24)(22,23,28,26);(5,9,10,6)(7,12,8,11)(13,15)(14,18)(16,20)(17,19)(21,26,27,23)(22,24,28,25)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=fgh,d^2=hi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329285, #72329343, #72329366
maximal normal subgroup of	#72329285, #72329343, #72329366
maximal quotient of	#72330223

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2235408);