#3364416

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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×#2404	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3428 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)(6,9)(7,11)(13,14,17,15)(16,18,20,19)(21,22)(23,27)(24,26)(25,28);(6,9)(10,12)(14,15)(18,19)(21,23,26,25)(22,27,24,28);(5,6)(7,10)(8,9)(11,12)(13,14,17,15)(16,19,20,18)(21,23,26,25)(22,28,24,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,18,20,19)(21,24,26,22)(23,27,25,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,25,26,23)(22,27,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293937, C₂×#3364416, #72326697, …
maximal normal subgroup of	#72293937, C₂×#3364416, #72326697, …
maximal quotient of	#72293980, C₂×#3364416

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3271612);