#3963401

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2408	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4110 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=gi,e^2=f,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	#72293907, C₂×#3963401, #72300946
maximal normal subgroup of	#72293907, C₂×#3963401, #72300946
maximal quotient of	#72293943, C₂×#3963401

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3870597);