#547024

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3334 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^2=ghi,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329279, #72329333, #72329370
maximal normal subgroup of	#72329279, #72329333, #72329370
maximal quotient of	#72329941

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 454220);