H(3,ℤ/8) #93252

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₄×C₈

Aliases

unitriangular	UT(3,ℤ/8)
Heisenberg	H(3,ℤ/8)

Subgroups

	subgroup	G/H
center, commutator	C₈	C₈²
Frattini, cosocle	#1726	C₂²
socle	C₂	#7014
prime radical	C₂³	H(3,ℤ/4)
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₈ ◁ G
LECS	G ▷ #1726 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

191 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,5,10)(3,13,15,38,17,40,45,63)(4,9,19,30)(6,11,24,35)(7,12,23,36)(8,25,26,52,28,53,56,43)(14,37,42,55,44,27,54,33)(16,39,22,50,41,61,51,62)(18,29,47,34)(20,31,49,59)(21,32,48,60)(46,57,64,58);(1,3)(2,8,11,27,12,28,29,37)(4,14)(5,22,23,51)(6,16)(7,17)(9,13,31,39,32,40,57,61)(10,33,34,26,36,55,35,56)(15,24,45,47)(18,41)(19,25,48,53)(20,43)(21,44)(30,50,58,38,60,62,59,63)(42,49,54,64)(46,52)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=h,f^2=i,b^a=bc,e^a=efi,h^a=hi,d^b=df,g^b=gi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210650, #72283827, #72283839, …
maximal normal subgroup of	#72210650, #72283827, #72283839, …
maximal quotient of	#72210629, #72210635, C₂×H(3,ℤ/8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 448);