#4430396

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3876 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326422, #72326423, #72326446
maximal normal subgroup of	#72326422, #72326423, #72326446

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4337592);