#4044698

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3450 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)(5,6,8,10)(7,11,12,9)(14,17)(15,19)(21,22)(23,25)(24,27)(26,28);(5,7,8,12)(6,9,10,11)(15,19)(18,20)(22,23)(24,28);(6,10)(9,11)(13,14)(15,18)(16,17)(19,20)(21,22,25,23)(24,27,28,26);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23,25,22)(24,26,28,27);(5,6)(7,11)(8,10)(9,12)(13,15)(14,18)(16,19)(17,20)(21,24)(22,26)(23,27)(25,28)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=h,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326687, #72326697, #72326752
maximal normal subgroup of	#72326687, #72326697, #72326752

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3951894);