#3859918

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
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
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3006 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)(8,10)(9,12)(13,14,18,20)(15,17,16,19)(22,24)(23,27);(5,6,8,10)(7,11,12,9)(13,15,18,16)(14,19,20,17)(23,27)(26,28);(5,6)(7,11)(8,10)(9,12)(13,15,18,16)(14,19,20,17)(21,22,25,24)(23,26,27,28);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,16,18,15)(14,17,20,19)(21,23,25,27)(22,26,24,28);(5,7,8,12)(6,9,10,11)(13,17)(14,16)(15,20)(18,19)(21,24,25,22)(23,28,27,26)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=g,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328344, #72328352, #72328449
maximal normal subgroup of	#72328344, #72328352, #72328449
maximal quotient of	#72330946, #72331620

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3767114);