#213848

order	512
nilpotency class	3
isoclinism	isoclinic class of #213828
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#35355	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1253 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,9,11,6)(4,12,13,7)(5,8)(10,14)(15,16)(17,18)(19,25,27,22)(20,28,29,23)(21,24)(26,31)(30,32);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,22)(20,26)(21,27)(23,31)(24,25)(28,30)(29,32);(6,9)(14,15)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,20)(18,23)(19,26)(21,29)(22,31)(24,28)(25,30)(27,32)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,g^c=gi,g^d=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240637, #72240638, #72240641, …
maximal normal subgroup of	#72240637, #72240638, #72240641, …
maximal quotient of	#72251640, #72251648

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121044);