#762858

order	512
nilpotency class	2
isoclinism	isoclinic class of #759288
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

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

4706 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)(5,6,10,8)(7,9,12,11)(13,14,16,18)(15,19,20,17)(21,22,24,26)(23,27,28,25);(5,7,10,12)(6,11,8,9)(13,14)(15,17)(16,18)(19,20)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(7,12)(9,11)(13,15,16,20)(14,17,18,19)(21,22)(23,27)(24,26)(25,28);(5,8,10,6)(7,9,12,11)(13,14)(15,17)(16,18)(19,20);(5,9)(6,7)(8,12)(10,11)(13,15)(14,17)(16,20)(18,19)(21,23)(22,25)(24,28)(26,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,e^c=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326453, #72326486, #72326509
maximal normal subgroup of	#72326453, #72326486, #72326509
maximal quotient of	#72328442

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 670054);