#1434840

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
automorphism	?
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
prime radical	#14716	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3255 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328543, #72328545, #72328639
maximal normal subgroup of	#72328543, #72328545, #72328639
maximal quotient of	#72329964

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1342036);