#642648

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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	C₂×#2847	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4144 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)(5,6,10,12)(7,9,8,11)(14,17)(18,20)(21,22,25,26)(23,24,27,28);(5,7,10,8)(6,11,12,9)(13,14)(15,18)(16,17)(19,20)(21,22)(23,24)(25,26)(27,28);(1,2,3,4)(6,12)(7,8)(15,19)(18,20)(22,26)(23,27);(5,8,10,7)(6,9,12,11)(13,15)(14,18)(16,19)(17,20)(21,23)(22,24)(25,27)(26,28);(5,9)(6,7)(8,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,28)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326456, #72326486, #72326506
maximal normal subgroup of	#72326456, #72326486, #72326506
maximal quotient of	#72328438

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 549844);